Dyons in higher-dimensional gauge theories

We discuss the ’t Hooft-Polyakov (TP) monopole and then dyon in the framework of higher dimensional gauge theories, such as gauge-Higgs unification models. First, we point out that the Bogomol’nyi-Prasad-Sommerfield (BPS) monopole is nothing but a self-dual gauge field in the 4-dimensional (4D) space including the extra dimension, which is argued to lead to a consequence that the mass of the BPS monopole <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>M</a:mi><a:mrow><a:mi>TPM</a:mi></a:mrow></a:msub></a:math> and therefore the vacuum expectation value (VEV) of the Higgs field are topologically quantized. In literatures, there exist related arguments on the calorons, which may be understood to be a composition of a pair of constituent monopole and antimonopole, with each constituent carrying fractional topological charge, while the net topological charge carried by the caloron is unity. From the viewpoint of the caloron, our conclusion of the quantized monopole mass corresponds to the special case, where only a single monopole exists that carries the net topological charge. Next, the argument is generalized to the case of the dyon. The mass of the BPS dyon, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>M</c:mi><c:mrow><c:mi>BPS</c:mi></c:mrow></c:msub></c:math>, is still proportional to the quantized Higgs VEV, though it also depends on a parameter <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>μ</e:mi></e:math>, denoting the ratio of the electric and magnetic charges of the dyon. In the 5D gauge theories the Chern-Simons term is induced at the quantum level, which, after the extra space component of the gauge field is replaced by its VEV, produces the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>θ</g:mi></g:math> term. Then, through the Witten effect we reach to an interesting conclusion that the parameter <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>μ</i:mi></i:math> and therefore <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>M</k:mi><k:mrow><k:mi>BPS</k:mi></k:mrow></k:msub></k:math> are discretized. In addition, we propose a numerical method to obtain the field configurations and the mass of the non-BPS dyons by use of “modified” gradient flow equations.